 Solution

ID: past-exam-of-the-martingales-and-financial-mathematics-course-of-the-university-of-oxford/2010/a/3/a/solution

The Borel-Cantelli second lemma says that independent events whose probabilities have divergent sum occur infinitely often almost surely. The events here are independent and have probabilities

min {1, λ / n}

. Their sum diverges by the harmonic series, so

P (U_{n} \leq λ / n infinitely often) = 1 (λ > 0) .

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